Since the first lunar missions in the 1960s, the moon has been the object of interest of both scientific research and potential commercial development. During the 1980s, several lunar missions were launched by national space agencies. Interest in the moon is increasing with the advent of the multi-national space station making it possible to stage lunar missions from low earth orbit. However, continued interest in the moon and the feasibility of a lunar base depends, in part, on the ability to schedule frequent and economical lunar missions.
A typical lunar mission comprises the following steps. Initially, a spacecraft is launched from earth or low earth orbit with sufficient impulse per unit mass, or change in velocity, to place the spacecraft into an earth-to-moon orbit. Generally, this orbit is a substantially elliptic earth-relative orbit having an apogee selected to nearly match the radius of the moon's earth-relative orbit.
As the spacecraft approaches the moon, a change in velocity is provided to transfer the spacecraft from the earth-to-moon orbit to a moon-relative orbit. An additional change in velocity may then be provided to transfer-the spacecraft from the moon-relative orbit to the moon's surface if a moon landing is planned. When a return trip to the earth is desired, another change in velocity is provided which is sufficient to insert the spacecraft into a moon-to-earth orbit, for example, an orbit similar to the earth-to-moon orbit. Finally, as the spacecraft approaches the earth, a change in velocity is required to transfer the spacecraft from the moon-to-earth orbit to a low earth orbit or an earth return trajectory.
FIG. 1 is an illustration of an orbital system in accordance with a conventional lunar mission in a non-rotating coordinate system wherein the X-axis 10 and Y-axis 12 are in the plane defined by the moon's earth-relative orbit 36, and the Z-axis 18 is normal to the plane. In a typical lunar mission, the spacecraft is launched from earth 16 or low earth orbit 20, not necessarily circular, and provided with sufficient velocity to place the spacecraft into an earth-to-moon orbit 22.
Near the moon 14, a change in velocity is provided to reduce the spacecraft's moon-relative energy and transfer the spacecraft into a moon-relative orbit 24 which is not necessarily circular. An additional change in velocity is then provided to transfer the spacecraft from the moon-relative orbit 24 to the moon 14 by way of the moon landing trajectory 25. When an earth-return is desired, a change in velocity sufficient to place the spacecraft into a moon-to-earth orbit 26 is provided either directly at the moon's surface or through multiple impulses as in the descent to the moon's surface. Finally, near the earth 16, a change in velocity is provided to reduce the spacecraft's earth-relative energy and return the spacecraft to low earth orbit 20 or to earth 16 via the earth-return trajectory 27.
It is desired to design a trajectory that minimizes fuel consumption, and which can deliver the spacecraft to a specified orbit around the moon, within a specified amount of time-of-flight. Usually, this problem is solved by a Hohmann transfer and patched conics approach, which patches together solutions from the earth-object and moon-object two-body problems. This approach leads to trajectories that can be completed in a small number of days, but with a suboptimal fuel consumption.
Recently, advances have been made to obtain greater understanding of the three-body problem that considers gravity of the moon, and finding trajectories which can use less fuel than the Hohmann-transfer based trajectories. However, the three-body problem is chaotic and highly sensitive to initial conditions. While above a minimum energy level, there are many trajectories the spacecraft can use. However, most of the trajectories take too long to be useful.
A typical trajectory in the three-body system is a spiraling trajectory. This feature is characteristic of chaotic systems. Hence the problem of finding the trajectories to the moon-orbit is a non-trivial task.
The conventional methods for determining trajectories as a three-body control problem have some important drawbacks. For example, methods based on weak-stability boundary (WSB) or methods based on bi-circular model transport the object very far away from earth (around 1.2 million K ms), which is undesirable due to limited capability of some ground-stations to monitor the object beyond the orbit of the moon.
Another method have computed trajectories from a very large earth orbit, and hence used manifold transfers directly. Also, those methods concentrate on finding specific trajectories, and are not sufficient to design end-to-end control procedure.
Another method directly transfers the object onto stable manifolds, but lacks the flexibility to satisfy various orbit constraints.
Accordingly, there is a need for a method that can systematically design low energy end-to-end trajectories from an orbit around the earth to an orbit around the moon.